Start with the governing idea: local curvature makes a disturbance evolve, and spatial coupling carries it onward. The familiar classical equation is the root. Its travelling-wave method then opens into four physical families—light, sound, water, and matter—each distinguished by what is waving and by its dispersion relation \(\omega(k)\).
HONESTY TIER A · DIRECT (water is B · linear theory) light and sound directly match the classical form; water and matter branch to their own evolution laws
› Waves & Oscillations › The Wave Equation & Four Families
The equation above is the classical archetype. It says that the acceleration of a field \(\Psi\) at one place is set by how sharply that field bends across neighbouring places. Put in a travelling pattern \(\Psi\propto\cos(kx-\omega t)\), and the differential equation becomes a relation between frequency and wavenumber. For the canonical equation that relation is \(\omega=vk\). The four branches below keep this same travelling-wave logic while changing the physical quantity and, in two cases, the evolution law itself.
The honest split: electromagnetic waves in vacuum and ideal sound are direct children of the second-order classical wave equation. Surface-water waves come from fluid motion plus free-surface boundary conditions; matter waves follow the first-order-in-time Schrödinger equation. What unifies all four is the plane-wave method and the resulting dispersion relation \(\omega(k)\). Here \(k=2\pi/\lambda\), \(\omega=2\pi f\), the crest speed is \(v_p=\omega/k\), and the packet speed is \(v_g=\mathrm{d}\omega/\mathrm{d}k\). Straight \(\omega(k)\) means a shape-preserving, non-dispersive wave; curved \(\omega(k)\) means dispersion and packet spreading.
Nothing material at all. A changing electric field \(\mathbf{E}\) makes a magnetic field \(\mathbf{B}\), whose change remakes \(\mathbf{E}\) — the two fields hold each other up and march through empty space. No medium, no ether.
Its geometry
\(\mathbf{E}\) and \(\mathbf{B}\) are transverse — both perpendicular to the travel direction and to each other, oscillating in step. Their ratio is fixed: \(|\mathbf{E}|=c|\mathbf{B}|\).
Its law
Non-dispersive: \(\omega=ck\) exactly, so every colour travels at the same \(c=299{,}792{,}458\ \mathrm{m/s}\) in vacuum. A pulse of light keeps its shape forever. Phase speed equals group speed equals \(c\).
λ ∝ 1.00 · c=λf = const
From Maxwell's equations in vacuum, each field component obeys the wave equation, and a plane wave running along \(x\) with the electric field along \(\hat{\mathbf{y}}\) is
$$\frac{\partial^2 \mathbf{E}}{\partial t^2}=c^2\nabla^2\mathbf{E},\qquad \mathbf{E}=E_0\cos(kx-\omega t)\,\hat{\mathbf{y}},\qquad \mathbf{B}=\frac{E_0}{c}\cos(kx-\omega t)\,\hat{\mathbf{z}},$$
with the dispersion relation \(\omega=ck\) and \(c=1/\sqrt{\mu_0\varepsilon_0}\). Because \(\omega\) is exactly proportional to \(k\), \(v_p=\omega/k=c\) and \(v_g=\mathrm{d}\omega/\mathrm{d}k=c\): light is the purest non-dispersive wave there is. Turn the frequency knob and the wavelength shortens, but the crest speed never changes.
The air itself. Each parcel of gas is pushed a little way along the travel direction and springs back — passing squeezes (compressions) and stretches (rarefactions) to its neighbour. What travels is the disturbance, not the air.
Its geometry
Longitudinal: the oscillation is along the direction of travel, not across it. There is nothing to polarise. Density and pressure ride a quarter-cycle out of step with the displacement.
Its law
Non-dispersive to excellent approximation: \(\omega=c_s k\) with \(c_s=\sqrt{\gamma P/\rho}\approx343\ \mathrm{m/s}\) in air at 20 °C. Music arrives as written because bass and treble travel together.
cs = 343 m/s · λ ∝ 1.00
A sound wave is a longitudinal displacement \(\xi(x,t)\) of the medium, whose gradient sets the pressure fluctuation:
$$\xi(x,t)=\xi_0\cos(kx-\omega t),\qquad \delta p=-K\frac{\partial\xi}{\partial x}=K\,\xi_0 k\,\sin(kx-\omega t),\qquad \omega=c_s k,\ \ c_s=\sqrt{\frac{\gamma P}{\rho}}.$$
The bulk modulus \(K\) and density \(\rho\) fix the speed; \(\gamma\) is the adiabatic index (1.4 for air, because compressions happen too fast to shed heat). The dots below are gas parcels displaced by \(\xi\) — watch them bunch into moving compressions while each stays near home. Pressure (the curve) peaks where the parcels are densest, a quarter wavelength from the points of largest displacement.
The height of the water surface, restored by gravity. But a parcel of water does not travel with the wave — it runs in a small circle, forward under a crest and backward under a trough, returning almost to where it started. The orbit shrinks with depth.
Its geometry
Neither purely transverse nor longitudinal — the motion is orbital, a blend of both. In deep water the circles decay as \(e^{ky}\); a few wavelengths down, the water barely stirs.
Its law
Dispersive: \(\omega=\sqrt{gk}\), so long waves travel faster than short ones. That is why a storm's chaos sorts itself into a clean, long-period swell by the time it reaches shore — and why crests race through a wave group at twice the group's speed: \(v_p=2v_g\).
depth:
vp = — · vg = —
Small-amplitude (linear, or "Airy") surface gravity waves obey, for a fluid depth \(h\) and surface tension neglected,
$$\omega^2=gk\,\tanh(kh)\ \ \xrightarrow[\text{deep }kh\gg1]{}\ \ \omega=\sqrt{gk},\qquad v_p=\sqrt{\frac{g}{k}},\quad v_g=\frac12 v_p.$$
A parcel at mean depth \(y<0\) traces a circle of radius \(a\,e^{ky}\), so the surface orbit (radius \(a\)) fades to nothing below. Because \(v_g=\tfrac12 v_p\) in deep water, a wave group advances at half the speed of the crests inside it: individual crests are born at the back of the group, sprint forward, and die at the front. Switch to shallow water (\(kh\ll1\)) and the law collapses to \(\omega=\sqrt{gh}\,k\) — a straight line again, non-dispersive, which is why tsunamis and sound-in-a-canal keep their shape. This is linearised theory: real steep, breaking waves need the full nonlinear equations, so this scene is tiered B.
A complex probability amplitude \(\psi\). There is no substance oscillating in space — \(|\psi|^2\) is the odds of finding the particle at each point. De Broglie's leap: every particle of momentum \(p\) has a wavelength \(\lambda=h/p\).
Its geometry
\(\psi\) is complex — an amplitude and a phase at every point, drawn here as a hue. The particle's speed is the speed of the packet (group), not of the coloured phase inside it.
Its law
Dispersive: \(\omega=\hbar k^2/2m\), a parabola. Now \(v_g=\hbar k/m=p/m\) is exactly the particle's velocity, while \(v_p=\hbar k/2m\) is only half of it — and because different \(k\) travel at different speeds, the packet inevitably spreads. A localised electron cannot stay localised.
vp = — · vg = — = 2vp
A free particle obeys the Schrödinger equation; a Gaussian bundle of plane waves \(e^{i(kx-\omega t)}\) weighted around \(k_0\) is summed live to build the packet:
$$i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}\ \Rightarrow\ \omega(k)=\frac{\hbar k^2}{2m},\qquad v_p=\frac{\omega}{k}=\frac{\hbar k}{2m},\quad v_g=\frac{\mathrm{d}\omega}{\mathrm{d}k}=\frac{\hbar k}{m}.$$
The envelope (probability) moves at \(v_g=p/m\), the true particle speed; the coloured carrier phase drifts backward through it at half that rate. Because \(\omega\) is curved, the packet's width grows without bound — the honest signature of the uncertainty principle in motion. Everything here is the exact free-particle solution (\(\hbar=m=1\) units); no potential, no approximation.
SYNTHESIS · ONE LAW
One Law, Four Waves — the Dispersion Diagram
TIER A · DIRECT
Read the slope
Draw \(\omega\) against \(k\). The chord from the origin to a point has slope \(\omega/k=v_p\) — the phase (crest) speed. The tangent at that point has slope \(\mathrm{d}\omega/\mathrm{d}k=v_g\) — the group (energy) speed.
Straight vs curved
Light and sound are straight lines through the origin: chord = tangent, so \(v_p=v_g\), pulses hold shape. Water (\(\sqrt{k}\)) bends down: \(v_gv_p\).
Drag the marker
Slide \(k\) and read all four \(v_p\) and \(v_g\) at once. The chord and tangent are drawn on each curve so you can literally see which waves keep their pulses and which smear.
show:
Slide the marker to compare the phase and group velocities of all four waves at a common wavenumber.
One travelling form, four governing laws
The shared skeleton is not one identical differential equation in every case; it is the plane-wave method. Write a field as a travelling phase pattern \(\cos(kx-\omega t)\) or \(e^{i(kx-\omega t)}\), substitute it into the appropriate governing law, and the result is a dispersion relation \(\omega(k)\). That function records how each physical system stores, returns, and transports energy at every scale. Its shape determines whether a pulse holds together, whether crests move with the energy, and whether a packet spreads.
Where \(\omega(k)\) is a straight line through the origin, the medium treats all wavelengths alike: light (\(\omega=ck\)) and sound (\(\omega=c_s k\)) are non-dispersive, phase and group speeds coincide, and a sharp pulse stays sharp — which is exactly why optical fibre and honest acoustics are possible. Where \(\omega(k)\) curves, wavelengths travel at different speeds and packets rearrange. Deep-water waves bend the curve downward (\(\omega\propto\sqrt{k}\)), so long swells outrun short chop and the crests you watch move at twice the speed of the group that carries the energy. Matter waves bend it upward (\(\omega\propto k^2\)), so a confined electron's packet spreads without limit and its group velocity — the thing that actually carries the particle — is double its phase velocity. Four unrelated corners of physics, one diagram, one law. Learn to read \(\omega(k)\) and you have read every wave at once.
Property
Electromagnetic
Sound
Water (deep)
Matter
What oscillates
E & B fields
air pressure / displacement
surface height (orbits)
probability amplitude ψ
Needs a medium?
no (vacuum)
yes (a fluid/solid)
yes (a fluid surface)
no — ψ is not in a medium
Orientation
transverse (E⊥B⊥k)
longitudinal
orbital (mixed)
complex, no spatial vector
Governing equation
∂²E/∂t² = c²∇²E
∂²ξ/∂t² = cₛ²∂²ξ/∂x²
ω²=gk·tanh(kh)
iℏ∂ψ/∂t = −(ℏ²/2m)∂²ψ/∂x²
Dispersion ω(k)
ck
cₛk
√(gk)
ℏk²/2m
Phase speed vₚ
c
cₛ
√(g/k)
ℏk/2m
Group speed v_g
c (= vₚ)
cₛ (= vₚ)
½√(g/k) (= ½vₚ)
ℏk/m (= 2vₚ)
Dispersive?
no — pulses hold shape
no — music arrives intact
yes — swell sorts by λ
yes — packets spread
Boundary of validity
EM wave
The exact vacuum plane-wave solution of Maxwell's equations: E and B transverse, in phase, \(|E|=c|B|\), \(\omega=ck\). Drawn as a real vector field. In a medium light does disperse (that is why prisms work); this scene is the vacuum case.
✓ exact vacuum solution
Sound wave
The linear acoustic solution: longitudinal \(\xi\), pressure \(=-K\,\partial\xi/\partial x\), \(\omega=c_s k\). Air is very weakly dispersive, so the straight-line law is excellent across the audible band. Parcels shown at true displacement.
✓ linear acoustics
Water wave
Linearised (Airy) small-amplitude theory: orbital motion \(a\,e^{ky}\), \(\omega^2=gk\tanh(kh)\). Faithful for gentle swell; real steep and breaking waves are nonlinear (Stokes, cnoidal), and surface tension rules the shortest ripples. Tiered B.
△ linear regime only
Matter wave
The exact free-particle Schrödinger packet, summed from plane waves with \(\omega=\hbar k^2/2m\) (\(\hbar=m=1\) units). Spreading and \(v_g=2v_p\) are literal outputs. Add a potential and the story becomes the bound-state pages.
✓ exact free packet
References
d'Alembert, J. (1747). Recherches sur la courbe que forme une corde tendue mise en vibration. (the wave equation)
Maxwell, J. C. (1865). A Dynamical Theory of the Electromagnetic Field. Phil. Trans. R. Soc. 155, 459.
Airy, G. B. (1841). Tides and Waves. Encyclopaedia Metropolitana. (linear water-wave theory)
Stokes, G. G. (1847). On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441.
Rayleigh, Lord (1877). The Theory of Sound. Macmillan. (group velocity, acoustics)
de Broglie, L. (1924). Recherches sur la théorie des quanta. PhD thesis, Paris. (λ = h/p)
Schrödinger, E. (1926). Quantisierung als Eigenwertproblem. Ann. Phys. 384, 361.
Lighthill, J. (1978). Waves in Fluids. Cambridge University Press.
Whitham, G. B. (1974). Linear and Nonlinear Waves. Wiley. (dispersion, group velocity)
Feynman, R. P. (1963). The Feynman Lectures on Physics, Vol. I, chs. 47–51. Addison-Wesley.